Refer to Table 4.1 when answering the following question.

Table 4.1: Production Model’s Prediction for Per Capita GDP (US = 1)

Observed per
capita GDP

Predicted per capita
output:
𝑦
=
𝑘
𝛼

Burundi
0.02

0.19

Brazil
0.25

0.75

Switzerland
1.20

1.10

China
0.30

0.85

Spain
0.63

1.03

United Kingdom
0.75

1.06

India
0.10

0.50

Italy
0.68

1.06

Japan
0.70

0.99

South Africa
0.20

0.61

(Source: Penn World Tables 9.0)

What is the productivity of South Africa that reconciles the data in the second and third columns of Table 4.1?

Round your answer to the nearest hundredth.

Question

Refer to Table 4.1 when answering the following question.

Table 4.1: Production Model’s Prediction for Per Capita GDP (US = 1)

Observed per
capita GDP
	
Predicted per capita
output:  
𝑦
=
𝑘
𝛼

Burundi	
0.02
	
0.19

Brazil	
0.25
	
0.75

Switzerland	
1.20
	
1.10

China	
0.30
	
0.85

Spain	
0.63
	
1.03

United Kingdom	
0.75
	
1.06

India	
0.10
	
0.50

Italy	
0.68
	
1.06

Japan	
0.70
	
0.99

South Africa	
0.20
	
0.61

(Source: Penn World Tables 9.0)

What is the productivity of South Africa that reconciles the data in the second and third columns of Table 4.1?

Round your answer to the nearest hundredth.

BlackTom AI · Accepted Answer

Per-capita output here is modeled as y = A · k^α, so productivity A scales the predicted term k^α to match observed y.

1) Relevant concepts
- The production model in per-capita form: y = A·k^α, where y is observed output per person, k^α is the predicted per-capita output from capital (given in the table as "Predicted per capita output: y = k^α"), and A is total factor productivity (TFP) that reconciles the two. To find A for a country, divide observed y by the model prediction k^α.

2) Step-by-step reasoning
- For South Africa the table gives:
  observed per-capita GDP y_obs = 0.20
  predicted k^α = 0.61
- Solve A from y_obs = A·(k^α) so A = y_obs / (k^α).

3) Calculations (all intermediate steps)
- A = 0.20 / 0.61
- Compute the quotient: 0.20 ÷ 0.61 = 0.327868852459...
- Round to the nearest hundredth: 0.327868... ≈ 0.33

4) Why this is correct
- The model’s prediction column is exactly k^α. Multiplying that predicted value by A must yield the observed per-capita GDP. Thus dividing observed by predicted yields the unique A that reconciles the two columns. The arithmetic above gives that A ≈ 0.33.

5) Common incorrect approaches (why other answers would be wrong)
- Subtracting (0.61 − 0.20 = 0.41) would measure the gap, not the multiplicative productivity factor.
- Reversing the division (0.61 / 0.20 = 3.05) would give how many times larger the predicted k^α is than observed y, not the productivity A.
- Using the US normalization or other rows instead of South Africa’s values would be incorrect because A must be country-specific computed from that country’s observed and predicted numbers.

6) Conclusion
- The productivity (TFP) for South Africa that reconciles the observed and predicted per-capita outputs is 0.33, which matches the provided answer.

类似问题

The observed capital-to-labor ratio in China is higher than in Brazil. However, China's GDP per capita is lower than Brazil's. It must be that Brazil's TFP is [ Select ] lower than the same as higher than China's.

Using the same table, you know that TFP 𝐽 𝑎 𝑝 𝑎 𝑛 TFP 𝑈 𝐾 equals ______. Round your answer to the nearest hundredth.

Refer to the following table when answering the following questions. In the following calculations, you should assume 𝛼 = 1 3 . The TFP that reconciles predicted and observed GDP per capita in India is ______. Round your answer to the nearest hundredth.

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